Creating fractals

$Cover image for Creating fractals$

Title:

Creating fractals

Personal Author:

Stevens, Roger T., 1927-

Publication Information:

Hingham, MA : Charles River Media, 2005

Physical Description:

1v + 1 CD-ROM

ISBN:

9781584504238

General Note:

Accompanied by compact disc : CP 11071

Subject Term:

Fractals

Curves

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010134534	QA614.86 S73 2005	Open Access Book	Book	Searching... Unknown

Everything You'll Need to Create Thousands of Fractals!Fractals are the name given to certain types of iterated equations that produce very strange results and are capable of creating unusual and beautiful patterns. Creating Fractals describes the characteristics and mathematical background of fractals and shows the reader how the accompanying fractal-generating program is used to produce thousands of different kinds of fractals, to enlarge them, to color them, and to save them-- without any knowledge of computers or programming. The program works with any computer using Windows. In addition to producing artistic effects, the reader can gain an understanding of how each type of fractal is created and how it might be used to treat natural phenomena, e.g., the turbulence of liquids, the behavior of the stock market, and the compression of graphic images. Mathematical terminology is explained in elementary terms.

Chapter 1 Introduction	p. 1
"Monster" Curves	p. 3
Working with Fractals	p. 3
The Lorenz and Other Strange Attractors	p. 4
What You Can Do with L-Systems Fractals	p. 5
The Snowflake and Other von Koch Curves	p. 6
Peano Curves	p. 6
Generators with Different Length Line Segments	p. 6
The Hilbert Curve	p. 7
FASS Curves	p. 7
Trees	p. 7
Creating Your Own L-Systems Fractals	p. 7
Newton's Method	p. 8
Fractals with the Logistic Equation	p. 8
Mandelbrot and Julia Sets	p. 10
Working with Colors	p. 11
Curves from Trigonometric and Exponential Functions	p. 12
Fractals Using Orthogonal Functions	p. 12
Phoenix Curves	p. 12
The Mandela Fractal	p. 13
Pokorny Fractals	p. 13
Fractals Using Circles	p. 13
Barnsley Fractals	p. 14
Iterated Function Systems	p. 14
Midpoint Displacement Fractals	p. 14
Continuing on Your Own	p. 15
References	p. 15
Chapter 2 What Are Fractals?	p. 17
Iterated Functions	p. 18
How Are Fractals Used?	p. 20
Basic Considerations	p. 20
Fractal Dimensions	p. 20
References	p. 22
Chapter 3 The Lorenz and Other Strange Attractors	p. 23
Strange Attractors	p. 25
The Lorenz Attractor	p. 25
Runge-Kutta Integration	p. 27
Viewing the Lorenz Attractor	p. 28
Other Strange Attractors	p. 29
References	p. 31
Chapter 4 What You Can Do with L-System Fractals	p. 33
How L-Systems Works	p. 34
The Geometric Basis for L-Systems	p. 35
Symbols Used in the L-Systems Language	p. 35
Overview of the L-Systems Program	p. 36
More Complex Generator Schemes	p. 37
Recurrence in L-Systems	p. 39
Creating L-Systems Fractals	p. 39
References	p. 40
Chapter 5 The Snowflake and Other von Koch Curves	p. 41
Snowflake Curve	p. 42
Gosper Curve	p. 46
3-Segment Quadric von Koch Curve	p. 53
8-Segment Quadric von Koch Curve	p. 54
18-Segment Quadric von Koch Curve	p. 61
32-Segment Quadric von Koch Curve	p. 64
50-Segment Quadric von Koch Curve	p. 67
Hexagonal 8-Segment von Koch Curve	p. 70
Sierpinski Triangle	p. 74
Islands Curve	p. 78
Chapter 6 Peano Curves	p. 83
Original Peano Curve	p. 84
Modified Peano Curve	p. 87
Cesaro Triangle	p. 89
Modified Cesaro Triangle	p. 93
Polya Triangle	p. 95
Peano-Gosper Curve	p. 98
Harter-Heighway Dragon Curve	p. 102
References	p. 105
Chapter 7 Generators with Different Sized Line Segments	p. 107
Peano 7-Segment Snowflake	p. 108
Peano 13-Segment Snowflake	p. 112
von Koch Curve Using Complex Generator	p. 115
Fractal Dimensions	p. 119
Chapter 8 The Hilbert Curve	p. 121
Hilbert II Curve	p. 126
Using the Hilbert Curve for Display Data Storage	p. 129
References	p. 129
Chapter 9 Fass Curves	p. 131
FASS 1 Curve	p. 132
FASS 2 Curve	p. 135
FASS 3 Curve	p. 136
FASS 4 Curve	p. 138
FASS 5 Curve	p. 140
FASS 6 Curve	p. 143
FASS 7 Curve	p. 146
Chapter 10 Tress	p. 149
Real Trees	p. 150
Tree Drawing with L-Systems	p. 152
Second Tree	p. 154
Third Tree	p. 157
Fourth Tree	p. 159
Fifth Tree	p. 162
Sixth Tree	p. 165
Bush	p. 167
Randomness in Trees	p. 169
References	p. 172
Chapter 11 Creating Your Own L-System Fractals	p. 173
Creating an Initiator and Generator	p. 174
Using the Create L-Systems Curve Command	p. 174
More Complicated Fractals with L-Systems	p. 175
Chapter 12 Newton's Method	p. 177
Noninteger Exponents	p. 181
Chapter 13 What You Can Do with Mandelbrot-like and Julia-like Fractals	p. 183
Julia Sets	p. 184
The Mandelbrot Set as a Map of Julia Sets	p. 185
Expansion of the Display	p. 185
References	p. 186
Chapter 14 The Mandelbrot and Julia Sets	p. 187
Expanding the Mandelbrot Set	p. 191
Julia Sets	p. 193
The Mandelbrot Set as a Map of Julia Sets	p. 195
Number of Iterations	p. 195
Specifying Julia Set Parameters	p. 196
Hypnoteyes	p. 197
Binary Decomposition	p. 197
References	p. 199
Chapter 15 Working with Colors	p. 201
Coloring L-Systems Fractals	p. 202
Mandelbrot Colors	p. 203
Julia Colors	p. 203
Dragon Colors	p. 204
Phoenix and Phoenix 2 Colors	p. 204
Blue and Silver	p. 205
Random Colors	p. 205
Custom Colors	p. 206
Complex Colors	p. 207
Gradient Colors	p. 208
Binary Decomposition	p. 210
Newton Colors	p. 210
Lyapunov Colors	p. 210
References	p. 222
Chapter 16 Fractals with the Logistic Equation	p. 211
Bifurcation Diagrams	p. 213
Expansion of the Display	p. 214
"Period Three Implies Chaos"	p. 216
The Fiegenbaum Number	p. 216
Self-Squared Dragons	p. 217
Lyapunov Fractals	p. 219
References	p. 222
Chapter 17 Fractals Using Transcendental Functions	p. 223
Cosine Fractal	p. 225
Sine Fractal	p. 227
Hyperbolic Cosine Fractal	p. 228
Hyperbolic Sine Fractal	p. 229
Exponential Fractal	p. 230
Chapter 18 Fractals Using Orthogonal Polynomials	p. 233
Creating Fractals with Orthogonal Polynomials	p. 235
Bernoulli Fractals	p. 236
Chebyshev Polynomials	p. 237
Legendre Polynomials	p. 240
Laguerre Polynomials	p. 241
Hermite Polynomials	p. 242
References	p. 244
Chapter 19 Creating Your Own Second-Order to Seventh-Order Equations	p. 245
Chapter 20 Phoenix Curves	p. 249
Relationship of Mandelbrot-Like and Julia-Like Phoenix Curves	p. 250
Phoenix Fractal Coloring	p. 251
Chapter 21 The Mandela and Pokorny Fractals	p. 253
Coloring the Mandela Curve	p. 254
Pokorny Fractals	p. 255
Chapter 22 Fractals Using Circles	p. 259
Apollonian Packing of Circles	p. 260
Soddy's Formula	p. 261
Creating the Apollonian Circle Packing Fractal	p. 261
Inversion	p. 265
Pharaoh's Breastplate	p. 266
Self-Homographic Fractals	p. 266
References	p. 268
Chapter 23 Barnsley Fractals	p. 271
The First Barnsley Fractal	p. 272
The Second Barnsley Fractal	p. 273
The Third Barnsley Fractal	p. 274
The Barnsley Sierpinski Triangle	p. 275
References	p. 276
Chapter 24 Iterated Function Systems	p. 277
Affine Transformation	p. 278
The Collage Theorem	p. 283
Creating Your Own IFS Fractals	p. 284
References	p. 285
Chapter 25 Midpoint Displacement Fractals	p. 287
Midpoint Displacement	p. 288
Triangles with a Common Line	p. 290
Midpoint Displacement in the Fractal Program	p. 290
Coloring Mountains	p. 291
Random Number Considerations	p. 292
Further Considerations	p. 293
About the CD-ROM	p. 295
Index	p. 299

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